# Theory Inductive

theory Inductive
imports Complete_Lattices Ctr_Sugar
```(*  Title:      HOL/Inductive.thy
Author:     Markus Wenzel, TU Muenchen
*)

section ‹Knaster-Tarski Fixpoint Theorem and inductive definitions›

theory Inductive
imports Complete_Lattices Ctr_Sugar
keywords
"inductive" "coinductive" "inductive_cases" "inductive_simps" :: thy_defn and
"monos" and
"print_inductives" :: diag and
"old_rep_datatype" :: thy_goal and
"primrec" :: thy_defn
begin

subsection ‹Least fixed points›

context complete_lattice
begin

definition lfp :: "('a ⇒ 'a) ⇒ 'a"
where "lfp f = Inf {u. f u ≤ u}"

lemma lfp_lowerbound: "f A ≤ A ⟹ lfp f ≤ A"
unfolding lfp_def by (rule Inf_lower) simp

lemma lfp_greatest: "(⋀u. f u ≤ u ⟹ A ≤ u) ⟹ A ≤ lfp f"
unfolding lfp_def by (rule Inf_greatest) simp

end

lemma lfp_fixpoint:
assumes "mono f"
shows "f (lfp f) = lfp f"
unfolding lfp_def
proof (rule order_antisym)
let ?H = "{u. f u ≤ u}"
let ?a = "⨅?H"
show "f ?a ≤ ?a"
proof (rule Inf_greatest)
fix x
assume "x ∈ ?H"
then have "?a ≤ x" by (rule Inf_lower)
with ‹mono f› have "f ?a ≤ f x" ..
also from ‹x ∈ ?H› have "f x ≤ x" ..
finally show "f ?a ≤ x" .
qed
show "?a ≤ f ?a"
proof (rule Inf_lower)
from ‹mono f› and ‹f ?a ≤ ?a› have "f (f ?a) ≤ f ?a" ..
then show "f ?a ∈ ?H" ..
qed
qed

lemma lfp_unfold: "mono f ⟹ lfp f = f (lfp f)"
by (rule lfp_fixpoint [symmetric])

lemma lfp_const: "lfp (λx. t) = t"
by (rule lfp_unfold) (simp add: mono_def)

lemma lfp_eqI: "mono F ⟹ F x = x ⟹ (⋀z. F z = z ⟹ x ≤ z) ⟹ lfp F = x"
by (rule antisym) (simp_all add: lfp_lowerbound lfp_unfold[symmetric])

subsection ‹General induction rules for least fixed points›

lemma lfp_ordinal_induct [case_names mono step union]:
fixes f :: "'a::complete_lattice ⇒ 'a"
assumes mono: "mono f"
and P_f: "⋀S. P S ⟹ S ≤ lfp f ⟹ P (f S)"
and P_Union: "⋀M. ∀S∈M. P S ⟹ P (Sup M)"
shows "P (lfp f)"
proof -
let ?M = "{S. S ≤ lfp f ∧ P S}"
from P_Union have "P (Sup ?M)" by simp
also have "Sup ?M = lfp f"
proof (rule antisym)
show "Sup ?M ≤ lfp f"
by (blast intro: Sup_least)
then have "f (Sup ?M) ≤ f (lfp f)"
by (rule mono [THEN monoD])
then have "f (Sup ?M) ≤ lfp f"
using mono [THEN lfp_unfold] by simp
then have "f (Sup ?M) ∈ ?M"
using P_Union by simp (intro P_f Sup_least, auto)
then have "f (Sup ?M) ≤ Sup ?M"
by (rule Sup_upper)
then show "lfp f ≤ Sup ?M"
by (rule lfp_lowerbound)
qed
finally show ?thesis .
qed

theorem lfp_induct:
assumes mono: "mono f"
and ind: "f (inf (lfp f) P) ≤ P"
shows "lfp f ≤ P"
proof (induct rule: lfp_ordinal_induct)
case mono
show ?case by fact
next
case (step S)
then show ?case
by (intro order_trans[OF _ ind] monoD[OF mono]) auto
next
case (union M)
then show ?case
by (auto intro: Sup_least)
qed

lemma lfp_induct_set:
assumes lfp: "a ∈ lfp f"
and mono: "mono f"
and hyp: "⋀x. x ∈ f (lfp f ∩ {x. P x}) ⟹ P x"
shows "P a"
by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp]) (auto intro: hyp)

lemma lfp_ordinal_induct_set:
assumes mono: "mono f"
and P_f: "⋀S. P S ⟹ P (f S)"
and P_Union: "⋀M. ∀S∈M. P S ⟹ P (⋃M)"
shows "P (lfp f)"
using assms by (rule lfp_ordinal_induct)

text ‹Definition forms of ‹lfp_unfold› and ‹lfp_induct›, to control unfolding.›

lemma def_lfp_unfold: "h ≡ lfp f ⟹ mono f ⟹ h = f h"
by (auto intro!: lfp_unfold)

lemma def_lfp_induct: "A ≡ lfp f ⟹ mono f ⟹ f (inf A P) ≤ P ⟹ A ≤ P"
by (blast intro: lfp_induct)

lemma def_lfp_induct_set:
"A ≡ lfp f ⟹ mono f ⟹ a ∈ A ⟹ (⋀x. x ∈ f (A ∩ {x. P x}) ⟹ P x) ⟹ P a"
by (blast intro: lfp_induct_set)

text ‹Monotonicity of ‹lfp›!›
lemma lfp_mono: "(⋀Z. f Z ≤ g Z) ⟹ lfp f ≤ lfp g"
by (rule lfp_lowerbound [THEN lfp_greatest]) (blast intro: order_trans)

subsection ‹Greatest fixed points›

context complete_lattice
begin

definition gfp :: "('a ⇒ 'a) ⇒ 'a"
where "gfp f = Sup {u. u ≤ f u}"

lemma gfp_upperbound: "X ≤ f X ⟹ X ≤ gfp f"
by (auto simp add: gfp_def intro: Sup_upper)

lemma gfp_least: "(⋀u. u ≤ f u ⟹ u ≤ X) ⟹ gfp f ≤ X"
by (auto simp add: gfp_def intro: Sup_least)

end

lemma lfp_le_gfp: "mono f ⟹ lfp f ≤ gfp f"
by (rule gfp_upperbound) (simp add: lfp_fixpoint)

lemma gfp_fixpoint:
assumes "mono f"
shows "f (gfp f) = gfp f"
unfolding gfp_def
proof (rule order_antisym)
let ?H = "{u. u ≤ f u}"
let ?a = "⨆?H"
show "?a ≤ f ?a"
proof (rule Sup_least)
fix x
assume "x ∈ ?H"
then have "x ≤ f x" ..
also from ‹x ∈ ?H› have "x ≤ ?a" by (rule Sup_upper)
with ‹mono f› have "f x ≤ f ?a" ..
finally show "x ≤ f ?a" .
qed
show "f ?a ≤ ?a"
proof (rule Sup_upper)
from ‹mono f› and ‹?a ≤ f ?a› have "f ?a ≤ f (f ?a)" ..
then show "f ?a ∈ ?H" ..
qed
qed

lemma gfp_unfold: "mono f ⟹ gfp f = f (gfp f)"
by (rule gfp_fixpoint [symmetric])

lemma gfp_const: "gfp (λx. t) = t"
by (rule gfp_unfold) (simp add: mono_def)

lemma gfp_eqI: "mono F ⟹ F x = x ⟹ (⋀z. F z = z ⟹ z ≤ x) ⟹ gfp F = x"
by (rule antisym) (simp_all add: gfp_upperbound gfp_unfold[symmetric])

subsection ‹Coinduction rules for greatest fixed points›

text ‹Weak version.›
lemma weak_coinduct: "a ∈ X ⟹ X ⊆ f X ⟹ a ∈ gfp f"
by (rule gfp_upperbound [THEN subsetD]) auto

lemma weak_coinduct_image: "a ∈ X ⟹ g`X ⊆ f (g`X) ⟹ g a ∈ gfp f"
apply (erule gfp_upperbound [THEN subsetD])
apply (erule imageI)
done

lemma coinduct_lemma: "X ≤ f (sup X (gfp f)) ⟹ mono f ⟹ sup X (gfp f) ≤ f (sup X (gfp f))"
apply (frule gfp_unfold [THEN eq_refl])
apply (drule mono_sup)
apply (rule le_supI)
apply assumption
apply (rule order_trans)
apply (rule order_trans)
apply assumption
apply (rule sup_ge2)
apply assumption
done

text ‹Strong version, thanks to Coen and Frost.›
lemma coinduct_set: "mono f ⟹ a ∈ X ⟹ X ⊆ f (X ∪ gfp f) ⟹ a ∈ gfp f"
by (rule weak_coinduct[rotated], rule coinduct_lemma) blast+

lemma gfp_fun_UnI2: "mono f ⟹ a ∈ gfp f ⟹ a ∈ f (X ∪ gfp f)"
by (blast dest: gfp_fixpoint mono_Un)

lemma gfp_ordinal_induct[case_names mono step union]:
fixes f :: "'a::complete_lattice ⇒ 'a"
assumes mono: "mono f"
and P_f: "⋀S. P S ⟹ gfp f ≤ S ⟹ P (f S)"
and P_Union: "⋀M. ∀S∈M. P S ⟹ P (Inf M)"
shows "P (gfp f)"
proof -
let ?M = "{S. gfp f ≤ S ∧ P S}"
from P_Union have "P (Inf ?M)" by simp
also have "Inf ?M = gfp f"
proof (rule antisym)
show "gfp f ≤ Inf ?M"
by (blast intro: Inf_greatest)
then have "f (gfp f) ≤ f (Inf ?M)"
by (rule mono [THEN monoD])
then have "gfp f ≤ f (Inf ?M)"
using mono [THEN gfp_unfold] by simp
then have "f (Inf ?M) ∈ ?M"
using P_Union by simp (intro P_f Inf_greatest, auto)
then have "Inf ?M ≤ f (Inf ?M)"
by (rule Inf_lower)
then show "Inf ?M ≤ gfp f"
by (rule gfp_upperbound)
qed
finally show ?thesis .
qed

lemma coinduct:
assumes mono: "mono f"
and ind: "X ≤ f (sup X (gfp f))"
shows "X ≤ gfp f"
proof (induct rule: gfp_ordinal_induct)
case mono
then show ?case by fact
next
case (step S)
then show ?case
by (intro order_trans[OF ind _] monoD[OF mono]) auto
next
case (union M)
then show ?case
by (auto intro: mono Inf_greatest)
qed

subsection ‹Even Stronger Coinduction Rule, by Martin Coen›

text ‹Weakens the condition \<^term>‹X ⊆ f X› to one expressed using both
\<^term>‹lfp› and \<^term>‹gfp››
lemma coinduct3_mono_lemma: "mono f ⟹ mono (λx. f x ∪ X ∪ B)"
by (iprover intro: subset_refl monoI Un_mono monoD)

lemma coinduct3_lemma:
"X ⊆ f (lfp (λx. f x ∪ X ∪ gfp f)) ⟹ mono f ⟹
lfp (λx. f x ∪ X ∪ gfp f) ⊆ f (lfp (λx. f x ∪ X ∪ gfp f))"
apply (rule subset_trans)
apply (erule coinduct3_mono_lemma [THEN lfp_unfold [THEN eq_refl]])
apply (rule Un_least [THEN Un_least])
apply (rule subset_refl, assumption)
apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
apply (rule monoD, assumption)
apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
done

lemma coinduct3: "mono f ⟹ a ∈ X ⟹ X ⊆ f (lfp (λx. f x ∪ X ∪ gfp f)) ⟹ a ∈ gfp f"
apply (rule coinduct3_lemma [THEN  weak_coinduct])
apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst])
apply simp_all
done

text  ‹Definition forms of ‹gfp_unfold› and ‹coinduct›, to control unfolding.›

lemma def_gfp_unfold: "A ≡ gfp f ⟹ mono f ⟹ A = f A"
by (auto intro!: gfp_unfold)

lemma def_coinduct: "A ≡ gfp f ⟹ mono f ⟹ X ≤ f (sup X A) ⟹ X ≤ A"
by (iprover intro!: coinduct)

lemma def_coinduct_set: "A ≡ gfp f ⟹ mono f ⟹ a ∈ X ⟹ X ⊆ f (X ∪ A) ⟹ a ∈ A"
by (auto intro!: coinduct_set)

lemma def_Collect_coinduct:
"A ≡ gfp (λw. Collect (P w)) ⟹ mono (λw. Collect (P w)) ⟹ a ∈ X ⟹
(⋀z. z ∈ X ⟹ P (X ∪ A) z) ⟹ a ∈ A"
by (erule def_coinduct_set) auto

lemma def_coinduct3: "A ≡ gfp f ⟹ mono f ⟹ a ∈ X ⟹ X ⊆ f (lfp (λx. f x ∪ X ∪ A)) ⟹ a ∈ A"
by (auto intro!: coinduct3)

text ‹Monotonicity of \<^term>‹gfp›!›
lemma gfp_mono: "(⋀Z. f Z ≤ g Z) ⟹ gfp f ≤ gfp g"
by (rule gfp_upperbound [THEN gfp_least]) (blast intro: order_trans)

subsection ‹Rules for fixed point calculus›

lemma lfp_rolling:
assumes "mono g" "mono f"
shows "g (lfp (λx. f (g x))) = lfp (λx. g (f x))"
proof (rule antisym)
have *: "mono (λx. f (g x))"
using assms by (auto simp: mono_def)
show "lfp (λx. g (f x)) ≤ g (lfp (λx. f (g x)))"
by (rule lfp_lowerbound) (simp add: lfp_unfold[OF *, symmetric])
show "g (lfp (λx. f (g x))) ≤ lfp (λx. g (f x))"
proof (rule lfp_greatest)
fix u
assume u: "g (f u) ≤ u"
then have "g (lfp (λx. f (g x))) ≤ g (f u)"
by (intro assms[THEN monoD] lfp_lowerbound)
with u show "g (lfp (λx. f (g x))) ≤ u"
by auto
qed
qed

lemma lfp_lfp:
assumes f: "⋀x y w z. x ≤ y ⟹ w ≤ z ⟹ f x w ≤ f y z"
shows "lfp (λx. lfp (f x)) = lfp (λx. f x x)"
proof (rule antisym)
have *: "mono (λx. f x x)"
by (blast intro: monoI f)
show "lfp (λx. lfp (f x)) ≤ lfp (λx. f x x)"
by (intro lfp_lowerbound) (simp add: lfp_unfold[OF *, symmetric])
show "lfp (λx. lfp (f x)) ≥ lfp (λx. f x x)" (is "?F ≥ _")
proof (intro lfp_lowerbound)
have *: "?F = lfp (f ?F)"
by (rule lfp_unfold) (blast intro: monoI lfp_mono f)
also have "… = f ?F (lfp (f ?F))"
by (rule lfp_unfold) (blast intro: monoI lfp_mono f)
finally show "f ?F ?F ≤ ?F"
qed
qed

lemma gfp_rolling:
assumes "mono g" "mono f"
shows "g (gfp (λx. f (g x))) = gfp (λx. g (f x))"
proof (rule antisym)
have *: "mono (λx. f (g x))"
using assms by (auto simp: mono_def)
show "g (gfp (λx. f (g x))) ≤ gfp (λx. g (f x))"
by (rule gfp_upperbound) (simp add: gfp_unfold[OF *, symmetric])
show "gfp (λx. g (f x)) ≤ g (gfp (λx. f (g x)))"
proof (rule gfp_least)
fix u
assume u: "u ≤ g (f u)"
then have "g (f u) ≤ g (gfp (λx. f (g x)))"
by (intro assms[THEN monoD] gfp_upperbound)
with u show "u ≤ g (gfp (λx. f (g x)))"
by auto
qed
qed

lemma gfp_gfp:
assumes f: "⋀x y w z. x ≤ y ⟹ w ≤ z ⟹ f x w ≤ f y z"
shows "gfp (λx. gfp (f x)) = gfp (λx. f x x)"
proof (rule antisym)
have *: "mono (λx. f x x)"
by (blast intro: monoI f)
show "gfp (λx. f x x) ≤ gfp (λx. gfp (f x))"
by (intro gfp_upperbound) (simp add: gfp_unfold[OF *, symmetric])
show "gfp (λx. gfp (f x)) ≤ gfp (λx. f x x)" (is "?F ≤ _")
proof (intro gfp_upperbound)
have *: "?F = gfp (f ?F)"
by (rule gfp_unfold) (blast intro: monoI gfp_mono f)
also have "… = f ?F (gfp (f ?F))"
by (rule gfp_unfold) (blast intro: monoI gfp_mono f)
finally show "?F ≤ f ?F ?F"
qed
qed

subsection ‹Inductive predicates and sets›

text ‹Package setup.›

lemmas basic_monos =
subset_refl imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
Collect_mono in_mono vimage_mono

lemma le_rel_bool_arg_iff: "X ≤ Y ⟷ X False ≤ Y False ∧ X True ≤ Y True"
unfolding le_fun_def le_bool_def using bool_induct by auto

lemma imp_conj_iff: "((P ⟶ Q) ∧ P) = (P ∧ Q)"
by blast

lemma meta_fun_cong: "P ≡ Q ⟹ P a ≡ Q a"
by auto

ML_file ‹Tools/inductive.ML›

lemmas [mono] =
imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
imp_mono not_mono
Ball_def Bex_def
induct_rulify_fallback

subsection ‹The Schroeder-Bernstein Theorem›

text ‹
▪ 🗏‹\$ISABELLE_HOME/src/HOL/ex/Set_Theory.thy›
▪ 🌐‹http://planetmath.org/proofofschroederbernsteintheoremusingtarskiknastertheorem›
▪ Springer LNCS 828 (cover page)
›

theorem Schroeder_Bernstein:
fixes f :: "'a ⇒ 'b" and g :: "'b ⇒ 'a"
and A :: "'a set" and B :: "'b set"
assumes inj1: "inj_on f A" and sub1: "f ` A ⊆ B"
and inj2: "inj_on g B" and sub2: "g ` B ⊆ A"
shows "∃h. bij_betw h A B"
proof (rule exI, rule bij_betw_imageI)
define X where "X = lfp (λX. A - (g ` (B - (f ` X))))"
define g' where "g' = the_inv_into (B - (f ` X)) g"
let ?h = "λz. if z ∈ X then f z else g' z"

have X: "X = A - (g ` (B - (f ` X)))"
unfolding X_def by (rule lfp_unfold) (blast intro: monoI)
then have X_compl: "A - X = g ` (B - (f ` X))"
using sub2 by blast

from inj2 have inj2': "inj_on g (B - (f ` X))"
by (rule inj_on_subset) auto
with X_compl have *: "g' ` (A - X) = B - (f ` X)"

from X have X_sub: "X ⊆ A" by auto
from X sub1 have fX_sub: "f ` X ⊆ B" by auto

show "?h ` A = B"
proof -
from X_sub have "?h ` A = ?h ` (X ∪ (A - X))" by auto
also have "… = ?h ` X ∪ ?h ` (A - X)" by (simp only: image_Un)
also have "?h ` X = f ` X" by auto
also from * have "?h ` (A - X) = B - (f ` X)" by auto
also from fX_sub have "f ` X ∪ (B - f ` X) = B" by blast
finally show ?thesis .
qed
show "inj_on ?h A"
proof -
from inj1 X_sub have on_X: "inj_on f X"
by (rule subset_inj_on)

have on_X_compl: "inj_on g' (A - X)"
unfolding g'_def X_compl
by (rule inj_on_the_inv_into) (rule inj2')

have impossible: False if eq: "f a = g' b" and a: "a ∈ X" and b: "b ∈ A - X" for a b
proof -
from a have fa: "f a ∈ f ` X" by (rule imageI)
from b have "g' b ∈ g' ` (A - X)" by (rule imageI)
with * have "g' b ∈ - (f ` X)" by simp
with eq fa show False by simp
qed

show ?thesis
proof (rule inj_onI)
fix a b
assume h: "?h a = ?h b"
assume "a ∈ A" and "b ∈ A"
then consider "a ∈ X" "b ∈ X" | "a ∈ A - X" "b ∈ A - X"
| "a ∈ X" "b ∈ A - X" | "a ∈ A - X" "b ∈ X"
by blast
then show "a = b"
proof cases
case 1
with h on_X show ?thesis by (simp add: inj_on_eq_iff)
next
case 2
with h on_X_compl show ?thesis by (simp add: inj_on_eq_iff)
next
case 3
with h impossible [of a b] have False by simp
then show ?thesis ..
next
case 4
with h impossible [of b a] have False by simp
then show ?thesis ..
qed
qed
qed
qed

subsection ‹Inductive datatypes and primitive recursion›

text ‹Package setup.›

ML_file ‹Tools/Old_Datatype/old_datatype_aux.ML›
ML_file ‹Tools/Old_Datatype/old_datatype_prop.ML›
ML_file ‹Tools/Old_Datatype/old_datatype_data.ML›
ML_file ‹Tools/Old_Datatype/old_rep_datatype.ML›
ML_file ‹Tools/Old_Datatype/old_datatype_codegen.ML›
ML_file ‹Tools/BNF/bnf_fp_rec_sugar_util.ML›
ML_file ‹Tools/Old_Datatype/old_primrec.ML›
ML_file ‹Tools/BNF/bnf_lfp_rec_sugar.ML›

text ‹Lambda-abstractions with pattern matching:›
syntax (ASCII)
"_lam_pats_syntax" :: "cases_syn ⇒ 'a ⇒ 'b"  ("(%_)" 10)
syntax
"_lam_pats_syntax" :: "cases_syn ⇒ 'a ⇒ 'b"  ("(λ_)" 10)
parse_translation ‹
let
fun fun_tr ctxt [cs] =
let
val x = Syntax.free (fst (Name.variant "x" (Term.declare_term_frees cs Name.context)));
val ft = Case_Translation.case_tr true ctxt [x, cs];
in lambda x ft end
in [(\<^syntax_const>‹_lam_pats_syntax›, fun_tr)] end
›

end
```